23. Prove that for any integer a, 9/ (a² – 3). -

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: 5. Let n Your-Reg-ID (7 digits number) and = let 21, 22, 23,..., Zn be the (distinct) n-th roots of…

A:

Q: Show that 1+2+3+… +n = (";").

A:

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Find the argument of 1+i ал with integer a,b b write your answer as a/b for Arg (1+i) =-

A: We have to find the argument of 1+i in the form of arg(1+i)=aπb with integer a, b.

Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: 61 Let A = Compute 612 - A and (612)A. 4 2

A: Step 1: Step 2: Step 3: Step 4:

Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

A:

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Let a1 = 1 and an+1 = 1 + 1/1 + an . Write out the first eight terms of {an}. Use part (a) to show…

A: a1=1 ,an+1=1+11+an

Q: Construct A so thatN(A) = all multiples of(4,3,2,l). Its rank is __ .

A: We have to construct a matrix A so that NA=all multiples of 4, 3, 2, 1 and also find its rank.

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

A:

Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: 2 4. Express 989 = 23 43 in the form x² + xy + 5y².

A: The given number is 989. The objective is to express the given number in the form of x2+xy+5y2.

Q: Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For…

A:

Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

A:

Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

A:

Q: (9) Prove that for integers x, y and z, if x² + y² = z² then either x or y is even.

A: We use method of contradiction to prove given statement.

Q: 49. Prove that (a – b) × (a + b) = 2(a × b).

A: The following properties of vectors are used in proving the given identity. a×a=b×b=0a×b=-(b×a)

Q: For all integers n>0, prove: 2n < (n+2)!

A:

Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

A: A perfect square is a positive integer whose square root is a positive integer. Example of perfect…